Unlike driving or taking the ferry, flying a pet into Great Britain isn't a free choice of any airline and any airport. The Animal and Plant Health Agency (APHA) maintains an approved list of airlines, airports, sea routes and rail carriers that are permitted to bring pets in under the pet travel scheme, and your pet has to travel on one of them.
A detail that genuinely catches people out: as of the current approved-routes list, there are no approved air routes into Wales at all. If your pet is flying rather than coming by Eurotunnel, ferry or car, the arrival point has to be an approved airport in England or Scotland, with onward travel into Wales handled separately by road.
Pets travelling from the Channel Islands, Isle of Man, Northern Ireland or the Republic of Ireland are exempt from the approved-route restriction. For everyone else, check your airline and arrival airport against the current APHA list before booking a flight, since an unapproved route simply won't be allowed to carry a pet, regardless of the airline's own general pet policy.